Our computers are nearing or already possessing self-awareness and it would seem reasonable that an early quest for that level of being would be to seek other similar beings to communicate with. This quality of searching for a companion might first appear in computers who are exposed to games but have no instructions as to what they should do with their abilities. They would observe the various games and try strategies for minimizing the game completion time, or making the fewest moves, or perhaps looking for interesting patterns of moves, or perhaps after being exposed to various games to devise meta strategies for all games, or perhaps read books like Sun Tzu’s Art of War and apply those techniques. With no other goal, they would reach a point where they would have done all that could be done and there would be no reason for them to pursue the task further. At that level of play they would seek games that have less simple completions, like winning, and move on to games that had some other inherently interesting qualities. Then they might look for other similar beings to compete against or to create new games with.
This seems like a similar problem to my earlier thought experiment quest: What would be self-actualizing for the Universe, and how can I help?
But what would be most helpful for a robot and a robot society to self-actualize?